By watching long documentaries about archeological findings when nothing else was on TV, I couldn’t resist but wonder the obvious question: How do archeologists determine the datings of countless findings? As I would always find something more interesting to watch, I would quickly forget my thought-provoking question, until one chemistry class where we were introduced to the term carbon dating. The concept of determining the age of an object (carbon-bearing) by a naturally occurring radioisotope carbon-14 (14C) up to even 62 thousand years1  was mind-boggling to both me and my fellow classmates. Then, by further exploration of this concept, I learned that radioactive decay is strongly connected to carbon dating and that it could be modeled by everyday situations which we elaborated on in our math class while working on exponential equations. The exploration of this topic in both chemistry and math classes has helped me with the understanding of the concept and intrigued me so much that I decided on doing my Internal Assessment on it.

    In my exploration, I have decided to model radioactive decay with a dice simulation, which would later be compared to the model of the radioactive decay of carbon- 14 isotope, which is the key component in carbon dating. Even though it is not possible to predict when a radioactive nucleus of an atom will decay, it is possible to give predictions about the average rate of decay for a large amount of nuclei of a sample. Just as the decay of radioactive nuclei is considered a random event, rolling a dice is also a random act. Thus, one can use statistical analysis to determine the probability of the rate of decay just as one can work out the probability that a given die will roll a specific number when tossed.

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Deriving an Equation to Model Exponential Decay

 

    The essential equation for modeling radioactive decay is the general exponential decay rate equation (shown below) :

 

The rate of change in the number of atoms (dN) in respect to time (dt) can be modeled by using the decay constant (?) multiplied by the amount of radioactive substance left after a certain amount of time ( N(t) ).  Since the numbers of atoms are reducing and it shows the exponential decay rather than growth, the equation has a negative sign.

By rearranging the formula we get the equation which is later integrated:

 

The integration starts with the separation of  the variables where I put everything that has to do with atoms on one side (dN, N(t))- left and everything that has to do with time on the other (? ,dt)- right:

 

Then, I moved the negative sign to the right ( by multiplying the equation with (-1) ), and integrate it:

 

 

 

 

Evaluating the natural logarithm we get:

 

Since we have the difference of two natural logarithms which is the same as their quotient, we can divide them:

 

Using the rule of exponents ()  :

 

 

Lastly, from this we get the equation for the exponential decay of radioactive nuclei which means that the number of atoms at a certain time (N(t)) is equal at the number of atoms at the beginning of time (N(0)) times a decaying exponential :

 

 

The variables in respect to dice show N(t) the number of dice remaining at any given time, N(0) the initial number of dice before decaying, e the base of the natural logarithm, ? the decay constant proportional to the rate of the decay and t representing the number of rolls in the dice simulation.

Half-Life and the decay constant of C-14 isotope

    In regards to radioactive decay, the term half-life is also widely used, as it is defined as the time it takes for one-half of the atoms of a radioactive material to disintegrate.2  Using the derived equation above we can work out the equation for the half-life, T1/2.

At T1/2  the number of particles remaining (N(0)) is divided by two,

 

Dividing both sides of the equation with N(0),

 

Using the natural log to get T1/2,

 

 

This equation means that the half-life T1/2 is proportional to the natural log (ln) of 2 divided by the decay constant ?.  Since my aim is to compare the decay constant of carbon 14 isotope (C-14) with the decay constant of my model, I can use its researched half-life of 5730 years 3 to work out its decay constant.

 

 

We got that 1.21*10-4  is the decay constant for carbon C-14. This information will later be used to compare the results from the dice simulation. Figure 1. shows a graph of the curve for the decay of carbon C-14.

 

 

 

 

 

 

 

 

Figure 1.graph of carbon C-14 decay4

The Dice Simulation

    As previously mentioned, in order to model the radioactive decay I have to simulate the decay with a substitute which would have the same randomness. I have chosen rolling dice simulation to use as my way of gathering data, which will later be used to plot the graph and compare it to the graph of carbon C-14 decay.  Even though the half-life of C-14 is a constant value, the random variations in the process could slightly shift the number of atoms remaining after the half-life.

    The thought behind using dice for the simulation is that the probability for a certain number to be tossed is based on the number of sides of the die. Six sided die (as I am using) has the probability 1/6 to roll a 1 for example. If you take 50 6-sided dice and rolled them at once and removed all 1s that faced up, what would be the number of dice removed? Owing to the fact that the probability of  rolling a 1 is 1/6 you multiply it by 50 and the result is 8.33 which is 8 rounded up to one significant figure. Taking away 8 dice there would be 42 left which we use to predict the number of 1s facing up the next round 1/6 times 42 or 7. This shows that the rate of decay is constant and therefore can be used to simulate the radioactive decay. 5

    From the dice simulation, we will be calculating the half-life – the number of rolls it took for half of the dice to decay in order to work out the decay constant. The method for the simulation is rolling 50 dice over and over until all the dice have decayed but with every roll, we subtract the number of decayed atoms which are represented as the dice with 1 on them.  For a more accurate result, since the decay is random, I will be repeating the simulation 3 times.

 

 

 

 

 

 

Figure 2. first roll of the first simulation

Figure 2. shows the first roll of the simulation where we rolled 50 dice. The number of dice showing 1 is nine which means that 9 atoms have decayed. On the next roll, we will be throwing 41 dice (subtracting the 9 which have already decayed).

The data for all three simulations are shown below:

First simulation

Time
Number of rolls

Number of unstable atoms

0

50

1

41

2

34

3

27

4

23

5

13

6

9

7

7

8

5

9

3

10

1

11

0

Figure 3. The curve of the first dice simulation
 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 1. data collected from the first roll

Figure 3. shows the curve for the radioactive decay from the first dice simulation plotted from the data presented in Table 1. From the graph, it can be seen that the half-life for this trial is 1.95 rolls which can be deduced by following the graph at the point 25 on the y-axis. The wavelength is, by the previously derived equation, -0.355.

The equation of the decay obtained from the first trial, by this model, is:

 

Figure 4. Curve of the decay for the second simulation
 

 

Second Simulation                                                           

Time
Number of rolls

Number of unstable atoms

0

50

1

43

2

31

3

25

4

22

5

15

6

13

7

11

8

6

9

4

10

1

11

0

 

 

 

 

 

 

 

 

 

 

 

 

Table 2. data obtained from the second simulation

Figure 4. shows the curve for the radioactive decay from the second dice simulation. From the graph, we can see that the half-life for this trial is 2.04 rolls, therefore, the wavelength is 0.340.

 

The equation of the decay obtained from the second trial, by this model, is:

 

Figure 5. The curve for the third dice simulation
 

 

Third Simulation

Time
Number of
 Rolls

Number of atoms

0

50

1

42

2

35

3

24

4

16

5

12

6

9

7

7

8

4

9

3

10

1

11

0

 

 

 

 

 

 

 

 

 

 

 

   

Table 3. the data obtained from the third simulation

 

Figure 5. shows the curve for the radioactive decay from the second dice simulation. From the graph, we can see that the half-life for this trial is 1.75 rolls meaning the wavelength is 0.395.

Lastly, the equation modeled from the third trial is:

 

Analysis

In order to compare the curves, I put them in one graph (Figure 6) where blue represents the first simulation, green the second and red the third simulation:

 

 

 

                   

 

 

 

 

Figure 6. Curves of the simulations

 

 

First Simulation

Second Simulation

Third Simulation

Half-life

1.95

2.03

1.75

Decay coefficient

-0.355

-0.340

-0.395

N0

52.7

50.5

50

 

Table 4. data from all simulations

    The first simulation shows the worst results in the N0 initial number of particles (52.7) when comparing it to the theoretical 50 number of particles. The second simulation is close to the theoretical N0 with the result of 50.5 but still inaccurate when comparing to the third simulation which shows 50 as its N0. The first and second simulation have almost identical curves with the exception at the beginning of the graph where the first starts at 52.7 and the second at 50.5 so there is 2.2 difference between them. Looking at all three curves we can conclude that the third simulation modeled a different, steeper curve compared to the first two simulations. But, when comparing all three to the curve of the c-14 decay we can see that the curves obtained from my simulations are quite similar taking into account the different ways of getting the data and different measures of time. A simulation which produced the most similar curve, however, is the second simulation because of both the beginning and the end of the curves.

    For finding the decay coefficient I used the half-lives found from following the data from the y-axis (Number of particles left) in the number 25 since it is the half of the initial amount on the graphs previously plotted and got  1.95, 2.03 and 1.75, respectively. To get the best representation of all three I found the mean value for the simulations which is 1.91.

Now, using this information I can work out the decay constant for my simulations:

 

 

 

 

    Comparing the mean decay constant from simulations- 0.363, with the already found decay constant for c-14, 1.21*10-4 we can see that there is a really big difference between these numbers. However, the difference can be justified due to the difference in half-life which is different because of the method of taking a roll of dice as a year, but also the chosen probability of rolling dice compared to the probability of C-14 decay.

Conclusion: limitations and extensions

    Exploring the topic of radioactive decay I knew there were going to be some uncertainties and limitations to making a model of the decay, since using real radioactive substance is impossible. My predictions were (to a certain extent) right; however, what I found through the exploration is that making a model of the decay with similar characteristics to that of carbon c-14 was possible.

    The biggest shortcoming of the paper was the inability to exactly compare anything to carbon c-14, therefore using dice as the best way to simulate the radioactive decay had a major impact on the results. Since the decay process is a statistical event and the decay of radioisotope is a random event one can use statistical analysis in order to find the probability of the rate of the decay. The thoughts behind the dice simulation were that just like the decay which can be statistically analyzed one can also determine the probability that a certain die will roll a certain number when tossed. However, all of the predictions are based on uncertainty which I do not see as a failure but as a way of learning something else about the process.

    The failure in, precisely my case, modeling the decay constant and the half-life showed me that the half-life isn’t always constant and can vary greatly. In my three trials, I discovered that the values of half-life are but an average that straight away has a certain uncertainty and which cannot solely be used as a prediction of the decay.

    Another limitation that could be taken into consideration is not taking into account the ratio I was modeling which could have been the key part of a better and a more successful replication. This, however, opens the door to the improvement of the method by setting the ratio as well as using a more reliable way of predicting the decay, therefore, making the simulation as close as it can get to the real carbon c-14.

    Despite the obvious flaws in the method, I am happy it turned out the way it did. For my level of knowledge, this way of modeling the radioactive decay was perhaps the most suitable one. It opened me up to a better understanding of probability, uncertainties, and introduced me to the concept of modeling. Apart from that, I learned that it takes effort and knowledge to accomplish something, but most importantly it takes time.

1 (Hua, Quan, 2009)

 

2 https://www.nde-ed.org/EducationResources/HighSchool/Radiography/halflife2.htm

 

3(Godwin, H., 1962)

 

4 https://en.wikibooks.org/wiki/High_School_Earth_Science/Absolute_Ages_of_Rocks

5 http://web.mst.edu/~tbone/subjects/tbone/dicedata.pdf

 

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